Born in Greenville, New York, in 1825, Ransome Bethune Welch completed his undergraduate studies at Union College, from which he graduated in 1846. After two years of theological studies at Andover Theological Seminary, he completed his divinity degree at Auburn Theological Seminary, from which he was graduated in 1852. He labored for the American Tract Society in Mississippi in 1853-54, and in the latter year was ordained by the Dutch Reformed Church in Gilboa, New York, where he served as pastor until 1856. He served the congregation at Catskill until 1859. In 1860, he was appointed professor of logic, rhetoric and English literature at Union College, where he taught until called in 1876 to Auburn Theological Seminary as professor and chair of Christian Theology.
Welch was prominent in the Presbyterian Church at the national and international levels, serving as a delegate to the Presbyterian Alliance at Belfast in 1884 and at London in 1888, and to the centennial conference of foreign missions at London in the latter year. He was the author of Faith and Modern Thought (1876) and Outlines of Christian Theology (1881), and in 1881 became an associate editor of the Presbyterian Review. His honors included Doctor of Divinity degrees from the University of the City of New York (now New York University) (1868) and Rutgers University (in the same year), and a Doctor of Laws degree from Maryville College (1872). Welch died on 29 June 1890 at Healing Spa, Virginia, and was interred in Auburn's Fort Hill Cemetery.
Memorial tributes to Welch were delivered by Prof. James Stevenson Riggs and others in the First Presbyterian Church on 11 November 1890. The addresses were published at Auburn in the following year.
Welch Memorial Hall, a classroom building on the campus of Auburn Theological Seminary, was completed in 1894 and named in his honor, as the construction of the classroom building and chapel was made possible by the professor's bequest to the seminary of $36,000. The gray and red stone building, connected to Willard Memorial Chapel, is one of the very few extant sites associated with the seminary, which relocated to New York City in 1939. Prior to the acquisition of the historic site by the Community Preservation Committee, Welch Memorial Building contained a white marble sculpture memorializing Welch by Herbert Adams and his wife Adeline Valentine Pond Adams. Regretably, the work of art was removed by Michael Dwyer prior to his sale of the building to the Community Preservation Committee. The 51 by 78-inch triptych relief sculpture, signed "Herbert Adams Adeline Adams MDCCCXCVIII," includes a waist-length portrait of Welch and bears the following inscription:
Ransome Bethune Welch DD LLD
Born January XXVII MDCCCXXIV
Died June XXIX DCCCXC
To give light to them that sit in darkness and in the shadow of death to guide our feet into the way of peace teaching every man in all wisdom that we may present every man perfect in Christ.
The January-February 1899 edition of The Auburn Seminary Review contained an article entitled "Unveiling of the Welch Memorial."